Lagrangian flows for vector fields with gradient given by a singular integral

TL;DR

This paper establishes quantitative estimates for flows of ODEs with vector fields whose gradients are singular integrals of L^1 functions, advancing the understanding of well-posedness beyond BV settings.

Contribution

It provides new quantitative stability, existence, and uniqueness results for flows with singular integral gradients, extending the classical BV framework.

Findings

Proves existence and uniqueness of flows with singular integral gradients.

Establishes quantitative stability and compactness results.

Extends well-posedness theory for continuity and transport equations.

Abstract

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an $L^1$ function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the $BV$ theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.